Problem 37 - Algebra 1: Concepts and Skills

Question

Simplify the expression if possible. $$ \frac{1-x}{x^{2}-x} $$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{1}{x}\).

Step 1: Analyze and rewrite the expression

Firstly, the expression is \(\frac{1-x}{x^{2}-x}\). The numerator is \(1-x\) and the denominator is \(x^{2}-x\). Upon examining the denominator \(x^{2}-x\), it can easily be factored since there is a common factor of \(x\) in all terms of the expression.

Step 2: Factorize the denominator

The denominator \(x^{2}-x\) can be rewritten as \(x(x-1)\), which is a result of factoring out the common factor. Therefore, the expression will now look like this: \(\frac{1-x}{x(x-1)}\).

Step 3: Simplify the expression

Looking at the expression again \(\frac{1-x}{x(x-1)}\), the numerator can be rewritten as \(-1*(x-1)\) because \(1-x\) is equivalent to \(-(x-1)\). This makes the expression look like this \(-1*\frac{x-1}{x(x-1)}\). Now \((x-1)\) in the numerator and \((x-1)\) in the denominator cancel out to give a simplified expression of \(-1*\frac{1}{x}\).

Key Concepts

Factoring Polynomials

The process of factoring polynomials is a crucial skill in algebra, which involves breaking down a complex expression into simpler parts that are multiplied together. Think of it as the reverse of multiplying polynomials; factoring is essentially finding the original expressions that were multiplied to give the polynomial you started with.

For instance, when you see an expression like \(x^2 - x\), factoring polynomials comes into play to simplify it. The goal is to identify common factors in all of the terms of the polynomial. In the given example, \(x\) is a common factor in both terms, so you can 'pull out' this common factor. The expression becomes \(x(x - 1)\). It’s like taking apart a puzzle you’ve put together — finding the pieces that, when combined, form the overall picture or, in this case, the polynomial.

Rational Expressions

In algebra, rational expressions are fractions that contain polynomials in the numerator, the denominator, or both. They're called 'rational' because they represent the ratio of two polynomials, just like a fraction represents the ratio of two numbers.

The process of simplifying rational expressions is akin to simplifying fractions; you want to reduce them to their simplest form. This often involves factoring the polynomials in the numerator and denominator and then canceling out any common factors. However, it's important to be aware that you can only cancel factors that are multiplied together, not those that are added or subtracted. Simplifying rational expressions reduces complexity and can make it easier to work with these expressions in equations or further algebraic manipulations.

Algebraic Fractions Simplification

Simplifying algebraic fractions, also known as algebraic fractions simplification, is a process similar to simplifying numerical fractions; you want to make the expression as clean and as easy to understand as possible. This involves several steps, including factoring polynomials and canceling common factors.

The key to successfully simplifying an algebraic fraction is to always look for common factors in the numerator and the denominator. Once you've factored these out, you can often cancel them, as they will form a ratio of one, effectively allowing you to 'simplify' the expression. Always remember that when simplifying, you are not changing the value of the expression, just its form. The simplified form is generally preferred, as it makes it easier to further manipulate or understand the expression.